Uniqueness of the viscosity solution of a constrained Hamilton-Jacobi equation

TL;DR

This paper proves the uniqueness of viscosity solutions for a constrained Hamilton-Jacobi equation relevant in quantitative genetics, under specific assumptions on the Hamiltonian and solution regularity.

Contribution

It provides a general proof of uniqueness for a class of constrained Hamilton-Jacobi equations, filling a gap in the mathematical theory.

Findings

Uniqueness established under convexity, monotonicity, and BV regularity assumptions.

Addresses a previously unresolved problem in the mathematical analysis of genetic models.

Extends the theoretical understanding of viscosity solutions in constrained PDEs.

Abstract

In quantitative genetics, viscosity solutions of Hamilton-Jacobi equations appear naturally in the asymptotic limit of selection-mutation models when the population variance vanishes. They have to be solved together with an unknown function I(t) that arises as the counterpart of a non-negativity constraint on the solution at each time. Although the uniqueness of viscosity solutions is known for many variants of Hamilton-Jacobi equations, the uniqueness for this particular type of constrained problem was not resolved, except in a few particular cases. Here, we provide a general answer to the uniqueness problem, based on three main assumptions: convexity of the Hamiltonian function H(I, x, p) with respect to p, monotonicity of H with respect to I, and BV regularity of I(t).